Chapter 11: Energy and its Conservation PHYSICS Principles and Problems

Chapter 11: Energy and its Conservation

PHYSICS Principles and

Problems

Within a closed and isolated system, energy can change form, but the total energy is constant.

CHAPTER

11 Energy and Its Conservation

BIG IDEA

Section 11.1 The Many Forms of Energy

Section 11.2 Conservation of Energy

CHAPTER

11 Table Of Contents

Click a hyperlink to view the corresponding slides. Exit

MAIN IDEA

Kinetic energy is energy due to an object’s motion, and potential energy is energy stored due to the interactions of two or more objects.

Essential Questions• How is a system’s motion related to its kinetic energy?

• What is gravitational potential energy?

• What is elastic potential energy?

• How are mass and energy related?

SECTION11.1

The Many Forms of Energy

Review Vocabulary

• system the object or objects of interest that can interact with each other and with the outside world

New Vocabulary• Rotational kinetic energy• Potential energy• Gravitational potential

energy

• Reference level• Elastic potential energy• Thermal energy

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• The word energy is used in many different ways in everyday speech.

• Some fruit-and-cereal bars are advertised as energy sources. Athletes use energy in sports.

• Companies that supply your home with electricity, natural gas, or heating fuel are called energy companies.

A Model of the Work-Energy Theorem

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• Scientists and engineers use the term energy much more precisely.

• Work causes a change in the energy of a system.

• That is, work transfers energy between a system and the external world.

A Model of the Work-Energy Theorem (cont.)

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• In this chapter, you will explore how objects can have energy in a variety of ways.

• Energy is like ice cream—it comes in different varieties. You can have vanilla, chocolate, or peach ice cream.

• They are different varieties, but they are all ice cream and serve the same purpose.

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• However, unlike ice cream, energy can be changed from one variety to another.

• In this chapter, you will learn how energy is transformed from one variety (or form) to another and how to keep track of the changes.

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• In the last chapter, you were introduced to the work-energy theorem.

• You learned that when work is done on a system, the energy of that system increases.

• On the other hand, if the system does work, then the energy of the system decreases.

• These are abstract ideas, but keeping track of energy is much like keeping track of your spending money.

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• If you have a job, the amount of money that you have increases every time you are paid. This process can be represented with a bar graph, as shown in the figure.

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• The orange bar represents how much money you had to start with, and the blue bar represents the amount that you were paid. The green bar is the total amount that you possess after the payment.

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• An accountant would say that your cash flow is positive.

• What happens when you spend the money that you have?

• The total amount of money that you have decreases.

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• As shown in the figure, the bar that represents the amount of money that you had before you bought that new CD is higher than the bar that represents the amount of money remaining after your shopping trip. The difference is the cost of the CD.

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• Cash flow is shown as a bar below the axis because it represents money going out, which can be shown as a negative number.

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• Energy is similar to your spending money.

• The amount of money that you have changes only when you earn more or spend it.

• Similarly, energy can be stored, and when energy is spent, it affects the motion of a system.

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• Gaining and losing energy also can be illustrated by throwing and catching a ball.

• You have already learned that when you exert a constant force, F, on an object through a distance, d, in the direction of the force, you do an amount of work, represented by W = Fd.

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• The work is positive because the force and motion are in the same direction, and the energy of the object increases by an amount equal to W.

• Suppose the object is a ball, and you exert a force to throw the ball.

• As a result of the force you apply, the ball gains kinetic energy.

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• The animation shows what happens when you throw the ball.

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• This time, the height of the bar represents the amount of work, or energy, measured in joules.

• The kinetic energy after the work is done is equal to the sum of the initial kinetic energy plus the work done on the ball.

• You can use a bar graph to represent the process.

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• What happens when you catch a ball?

• Before hitting your hands or glove, the ball is moving, so it has kinetic energy.

• In catching it, you exert a force on the ball in the direction opposite to its motion.

• Therefore, you do negative work on it, causing it to stop.

• Now that the ball is not moving, it has no kinetic energy.

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• The animation shows what happens when you catch the ball.

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• Kinetic energy is always positive, so the initial kinetic energy of the ball is positive.

• The work done on the ball is negative and the final kinetic energy is zero.

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• Again, the kinetic energy after the ball has stopped is equal to the sum of the initial kinetic energy plus the work done on the ball.

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Kinetic Energy

• The kinetic energy is proportional to the object’s mass.

• A 7.26-kg bowling ball thrown through the air has much more kinetic energy than a 0.148-kg baseball with the same velocity, because the bowling ball has a greater mass.

• Recall that translational kinetic energy, , where m is the mass of the object and v is the magnitude of its velocity.

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• The kinetic energy of an object is also proportional to the square of the object’s velocity.

• A car speeding at 20 m/s has four times the kinetic energy of the same car moving at 10 m/s.

• Kinetic energy also can be due to rotational motion.

Kinetic Energy (cont.)

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• If you spin a toy top in one spot, does it have kinetic energy?

• You might say that it does not because the top is not moving anywhere.

• However, to make the top rotate, someone had to do work on it.

• Therefore, the top has rotational kinetic energy. This is one of the several varieties of energy.

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• The kinetic energy of a spinning object is called rotational kinetic energy.

• Rotational kinetic energy can be calculated using; , where I is the object’s moment of inertia and ω is the object’s angular velocity.

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• A diver does work as she is diving off the diving board. This work produces both translational and rotational kinetic energies.

• When the diver’s center of mass moves as she leaps, translational kinetic energy is produced. When she rotates about her center of mass, rotational kinetic energy is produced.

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• Because she is moving toward the water and rotating at the same time while in the tuck position, she has both translational and rotational kinetic energy.

• When she slices into the water, she has translational kinetic energy.

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• Imagine a group of boulders high on a hill.

• These boulders have been lifted up by geological processes against the force of gravity; thus, they have stored energy.

• Energy that is stored due to interactions between objects in a system is called potential energy.

• In a rock slide, the boulders are shaken loose.

• They fall and pick up speed as their potential energy is converted to kinetic energy.

Potential Energy

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• In the same way, a small spring-loaded toy, such as a jack-in-the-box, has stored energy, but the energy is stored in a compressed spring.

• While both of these examples represent energy stored by mechanical means, there are many other means of storing energy.

Potential Energy (cont.)

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• Automobiles, for example, carry their energy stored in the form of chemical energy in the gasoline tank.

• Energy is made useful or causes motion when it changes from one form to another.

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• How does the money model that was discussed earlier, illustrate the transformation of energy from one form to another?

• Money, too, can come in different forms.

• You can have one five-dollar bill or 20 quarters or 500 pennies; in all of these cases, you still have five dollars.

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• The height of the bar graph in the figure represents the amount of money in each form.

• In the same way, you can use a bar graph to represent the amount of energy in various forms that a system has.

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Gravitational Potential Energy

Click image to view the movie.

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• In the equation for gravitational potential energy, g is the acceleration due to gravity. Gravitational potential energy, like kinetic energy, is measured in joules.

Gravitational Potential Energy (cont.)

Gravitational Potential Energy GPE = mgh

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• Consider the energy of a system consisting of an orange used by the juggler plus Earth.

• The energy in the system exists in two forms: kinetic energy and gravitational potential energy.

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• At the beginning of the orange’s flight, all the energy is in the form of kinetic energy, as shown in the figure.

• On the way up, as the orange slows down, energy changes from kinetic energy to potential energy.

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• At the highest point of the orange’s flight, the velocity is zero.

• Thus, all the energy is in the form of gravitational potential energy.

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• On the way back down, potential energy changes back into kinetic energy.

• The sum of kinetic energy and potential energy is constant at all times because no work is done on the system by any external forces.

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• The animation shows kinetic energy and potential energy of a system consisting of an orange used by the juggler plus Earth.

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• In the figure, the reference level is the dotted line, which represents the point at which the orange left the juggler’s hand.

• That is, the height of the orange is measured from the dotted line.

Reference Levels

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• Thus, at the dotted line, h = 0 m and PE = 0 J.

• You can set the reference level at any height that is convenient for solving a given problem.

Reference Levels (cont.)

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• The animation shows what happens when you set the reference level at the highest point of the orange’s flight.

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• When the reference level is set at the highest point of the orange’s flight, then h = 0 m and the system’s PE = 0 J at that point, as illustrated in the figure.

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• The potential energy of the system is negative at the beginning of the orange’s flight, zero at the highest point, and negative at the end of the orange’s flight.

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• If you were to calculate the total energy of the system represented by the figure on the left, it would be different from the total energy of the system represented by the figure on the right.

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• The total energy of the system would be different because the reference levels are different in each situation.

• However, the total energy of the system in each situation would be constant at all times during the flight of the orange.

• Only changes in energy determine the motion of a system.

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You lift a 7.30-kg bowling ball from the storage rack and hold it up to your shoulder. The storage rack is 0.610 m above the floor and your shoulder is 1.12 m above the floor.

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When the ball is at your shoulder…

A. What is the ball-Earth system’s gravitational potential energy relative to the floor?

B. What is the ball-Earth system’s gravitational potential energy relative to the storage rack?

C. How much work was done by gravity as you lifted the ball from the rack to shoulder level?

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Step 1: Analyze and Sketch the Problem

• Sketch the situation. • Choose a reference level.• Draw a bar graph showing

the gravitational potential energy with the floor as the reference level.

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Known:

m = 7.30 kg

hr = 0.610 m (rack relative to the floor)

hs = 1.12 m (shoulder relative to the floor)

g = 9.80 N/kg

Unknown:

GPEs rel f = ?

GPEs rel r = ?

Identify the known and unknown variables.

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Step 2: Solve for the Unknown

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a. Set the reference level to be at the floor.

Determine the gravitational potential energy of the ball at shoulder level.

Substitute m = 7.30 kg, g = 9.80 N/kg, hs rel f = 1.12 m

GPEs rel f = (7.30 kg)(9.80 N/kg)(1.12 m)

GPEs rel f = mghs

GPEs rel f = 8.01×101 J

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b.Set the reference level to be at the rack height.

Determine the height of your shoulder relative to the rack.

h = hs – hr

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Determine the gravitational potential energy of the system when the ball is at shoulder level.

GPEs rel r = mgh

Substitute h = hs – hr

PE = mg (hs – hr)

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Substitute m = 7.3 kg, g = 9.80 N/kg, hs = 1.12 m, hr = 0.610 m

GPE = (7.30 kg)(9.80 N/kg)(1.12 m – 0.610 m)

GPE = 36 J

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c. The work done by gravity is the weight of the ball times the distance the ball was lifted.

Because the weight opposes the motion of lifting, the work is negative.

W = Fd

W = – (mg)h

W = – (mg) (hs – hr)

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Substitute m = 7.30 kg, g = 9.80 N/kg, hs = 1.12 m, hr = 0.610 m

W = – (7.30 kg)(9.80 N/kg)(1.12 m – 0.610 m)

W = – 36 J

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Step 3: Evaluate the Answer

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Are the units correct?

The potential energy and work are both measured in joules.

Is the magnitude realistic?

The ball should have a greater potential energy relative to the floor than relative to the rack, because the ball’s distance above the floor is greater.

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The steps covered were:

Sketch the situation.

Choose a reference level.

Draw a bar graph showing the gravitational potential energy.

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• When a string on a bow is pulled, work is done on the bow, storing energy in it.

• Thus, the energy of the system increases.

• Identify the system as the bow, the arrow, and Earth.

• When the string and arrow are released, energy is changed into kinetic energy.

• The stored energy due to an object’s change in shape, like the pulled string, is called elastic potential energy, which is often stored in rubber balls, rubber bands, slingshots, and trampolines.

Elastic Potential Energy

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• Energy also can be stored in the bending of an object.

• When stiff metal or bamboo poles were used in pole-vaulting, the poles did not bend easily.

• Little work was done on the poles, and consequently, the poles did not store much potential energy.

• Since flexible fiberglass poles were introduced, however, record pole-vaulting heights have soared.

Elastic Potential Energy (cont.)

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• A pole-vaulter runs with a flexible pole and plants its end into the socket in the ground.

• When the pole-vaulter bends the pole some of the pole-vaulter’s kinetic energy is converted to elastic potential energy.

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• When the pole straightens, the elastic potential energy is converted to gravitational potential energy and kinetic energy as the pole-vaulter is lifted as high as 6 m above the ground.

• Unlike stiff metal poles or bamboo poles, fiberglass poles have an increased capacity for storing elastic potential energy.

• Thus, pole-vaulters are able to clear bars that are set very high.

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• Albert Einstein recognized yet another form of potential energy: mass itself.

• He said that mass, by its very nature, is energy.

• This energy, E0, is called rest energy and is represented by the following famous formula.

Rest Energy E0 = mc2

• The rest energy of an object is equal to the object’s mass times the speed of light squared.

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• According to this formula, stretching a spring or bending a vaulting pole causes the spring or pole to gain mass.

• In these cases, the change in mass is too small to be detected.

• When forces within the nucleus of an atom are involved, however, the energy released into other forms, such as kinetic energy, by changes in mass can be quite large.

Mass (cont.)

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• Think about all the form and sources of energy you encounter every day: – Chemical energy: released from the burning of fossil

fuels and during digestion.– Nuclear energy: released with the structure of an

atom’s nucleus changes.– Thermal energy: associated with temperature. It is

the sum of the kinetic energy and potential energy of the particles in a system.

– Electrical energy: associated with charged particles.

Other Forms of Energy

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A. His kinetic energy will be doubled.

B. His kinetic energy will be halved.

C. His kinetic energy will increase by a factor of four.

D. His kinetic energy will decrease by a factor of four.

A boy running on a track doubles his velocity. Which of the following statements about his kinetic energy is true?

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Section Check

Reason: The kinetic energy of an object is equal to half times the mass of the object multiplied by the speed of the object squared.

Kinetic energy is directly proportional to the square of velocity. Since the mass remains the same, if the velocity is doubled, his kinetic energy will increase by a factor of four.

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Section Check

Answer

If an object moves away from Earth, energy is stored in the system as the result of the force between the object and Earth. What is this stored energy called?

A. rotational kinetic energy

B. gravitational potential energy

C. elastic potential energy

D. linear kinetic energy

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Section Check

Reason: The energy stored due to the gravitational force between an object and Earth is called gravitational potential energy.

Gravitational potential energy is represented by the equation, PE = mgh. That is, the gravitational potential energy of an object is equal to the product of its mass, the acceleration due to gravity, and the height above the reference level. Gravitational potential energy is measured in joules (J).

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Section Check

Answer

Two girls, Sarah and Susan, with equal masses are jumping on a floor. If Sarah jumps higher, what can you say about the gain in their gravitational potential energies?

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Section Check

A. Since both have equal masses, they gain equal gravitational potential energies.

B. The gain in gravitational potential energy of Sarah is greater than that of Susan.

C. The gain in gravitational potential energy of Susan is greater than that of Sarah.

D. Neither Sarah nor Susan possesses gravitational potential energy.

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Section Check

Reason: The gravitational potential energy of an object is equal to the product of its mass, the acceleration due to gravity, and the distance from the reference level.

Gravitational potential energy is directly proportional to height. Because the masses of Sarah and Susan are the same, the one jumping to a greater height (Sarah) will gain greater potential energy.

PE = mgh

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Section Check

Answer

MAIN IDEA

In a collision in a closed, isolated system, the total energy is conserved, but kinetic energy might not be conserved.

Essential Questions• Under what conditions is energy conserved?

• What is mechanical energy, and when is it conserved?

• How are momentum and kinetic energy conserved or changed in a collision?

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Conservation of Energy

Review Vocabulary

• Closed system a system that does not gain or lose mass

New Vocabulary• Law of conservation of energy• Mechanical energy• Elastic collision• Inelastic collision

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• The money model can again be used to illustrate what is happening in conservation of energy.

• Suppose you have a total of $50 in cash. One day, you count your money and discover that you are $3 short.

• Would you assume that the money just disappeared?

Law of Conservation of Energy

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• You probably would try to remember whether you spent it, and you might even search for it. In other words, rather than giving up on the conservation of money, you would try to think of different places where it might have gone.

Law of Conservation of Energy (cont.)

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• Scientists do the same thing as you would if you could not account for a sum of money.

• Whenever they observe energy leaving a system, they look for new forms into which the energy could have been transferred.

• This is because the total amount of energy in a system remains constant as long as the system is closed and isolated from external forces.

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• The Law of conservation of energy states that in a closed, isolated system, energy can neither be created nor destroyed; rather, energy is conserved.

• Under these conditions, energy can change form but the system’s total energy in all of its forms remains constant.

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• For many situations you focus on the energy that comes from the motions of and interactions between objects.

• The sum of the kinetic energy and the potential energy of the objects in a system is the system’s mechanical energy (ME).

• Kinetic energy includes both the translational and rotational kinetic energies.

• Potential energy includes the gravitational and elastic potential energies.

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• Mechanical Energy of a System

ME = KE + PE

• The mechanical energy of a system is equal to the sum of the kinetic energy and potential energy if no other forms of energy are present.

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Click image to view the movie.

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• In the case of a roller coaster that is nearly at rest at the top of the first hill, the total mechanical energy in the system is the coaster’s gravitational potential energy at that point.

• Suppose some other hill along the track is higher than the first one.

• The roller coaster would not be able to climb the higher hill because the energy required to do so would be greater than the total mechanical energy of the system.

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• Suppose you ski down a steep slope.

• When you begin from rest at the top of the slope, your total mechanical energy is simply your gravitational potential energy.

• Once you start skiing downhill, your gravitational potential energy is converted to kinetic energy.

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• As you ski down the slope, your speed increases as more of your potential energy is converted to kinetic energy.

• In ski jumping, the height of the ramp determines the amount of energy that the jumper has to convert into kinetic energy at the beginning of his or her flight.

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• The simple oscillation of a pendulum also demonstrates conservation of energy.

• The system is the pendulum bob and Earth. Usually, the reference level is chosen to be the height of the bob at the lowest point, when it is at rest. If an external force pulls the bob to one side, the force does work that gives the system mechanical energy.

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• At the instant the bob is released, all the energy is in the form of potential energy, but as the bob swings downward, the energy is converted to kinetic energy.

• The figure shows a graph of the changing potential and kinetic energies of a pendulum.

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• When the bob is at the lowest point, its gravitational potential energy is zero, and its kinetic energy is equal to the total mechanical energy in the system.

• Note that the total mechanical energy of the system is constant if we assume that there is no friction.

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• The oscillations of a pendulum eventually come to a stop, a bouncing ball comes to rest, and the heights of roller coaster hills get lower and lower.

• Where does the mechanical energy in such systems go?

• Any object moving through the air experiences the forces of air resistance.

• In a roller coaster, there are frictional forces between the wheels and the tracks.

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• When a ball bounces off of a surface, all of the elastic potential energy that is stored in the deformed ball is not converted back into kinetic energy after the bounce.

• Some of the energy is converted into thermal energy and sound energy.

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Conservation of Mechanical EnergyDuring a hurricane, a large tree limb, with a mass of 22.0 kg and at a height of 13.3 m above the ground, falls on a roof that is 6.0 m above the ground.

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A. Ignoring air resistance, find the kinetic energy of the limb when it reaches the roof.

B. What is the speed of the limb when it reaches the roof?

Conservation of Mechanical Energy (cont.)

• Sketch the initial and final conditions.

• Choose a reference level.

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• Draw a bar graph.

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Identify the known and unknown variables.

Unknown:

GPEi = ? KEf = ?

GPEf = ? vf = ?

Known:

m = 22.0 kg g = 9.80 N/kg

hlimb = 13.3 m vi = 0.0 m/s

hroof = 6.0 m KEi = 0.0 J

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A. Set the reference level as the height of the roof. Solve for the initial height of the limb relative to the roof.

h = hlimb – hroof

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Substitute hlimb = 13.3 m, hroof = 6.0 m

h = 13.3 m – 6.0 m

= 7.3 m

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Solve for the initial potential energy of the limb-Earth system.

GPEi = mgh

Substitute m = 22.0 kg, g = 9.80 N/kg, h = 7.3 m

PEi = (22.0 kg) (9.80 N/kg) (7.3 m)

= 1.6×103 J

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The tree limb is initially at rest.

Identify the initial kinetic energy of the limb.

KEi = 0.0 J

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Identify the final potential energy of the system.

h = 0.0 m at the roof.

GPEf = 0.0 J

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Substitute KEi = 0.0 J, GPEi = 1.6 x 103 J and GPEi = 0.0 J.

Use the principle of conservation of mechanical energy to find the KEf.

KEf + GPEf = KEi + GPEi

KEf = (0.0 J) + (1.6×103 J) – (0.0 J)

= 1.6 x 103 J

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B. Solve for the speed of the limb.

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Substitute KEf = 1.6×103 J, m = 22.0 kg

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Are the units correct?

Velocity is measured in m/s and energy is measured in kg·m2/s2 = J.

Do the signs make sense?

KE and the magnitude of velocity are always positive.

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Sketch the initial and final conditions.

Choose a reference level.

Draw a bar graph.

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Set the reference level as the height of the roof. Solve for the initial height of the limb relative to the roof.

Solve for the speed of the limb.

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• A collision between two objects, whether the objects are automobiles, hockey players, or subatomic particles, is one of the most common situations analyzed in physics.

• Because the details of a collision can be very complex during the collision itself, the strategy is to find the motion of the objects just before and just after the collision.

Analyzing Collisions

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• What conservation laws can be used to analyze such a system?

• If the system is isolated, then momentum and energy are conserved.

• However, the potential energy or thermal energy in the system may decrease, remain the same, or increase.

• Therefore, you cannot predict whether kinetic energy is conserved.

Analyzing Collisions (cont.)

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• Consider the collision shown in the figure.

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• In Case 1, the momentum of the system before and after the collision is represented by the following:

pi = pCi+pDi = (1.00 kg)(1.00 m/s)+(1.00 kg)(0.00 m/s)

= 1.00 kg·m/s

pf = pCf+pDf = (1.00 kg)(–0.20 m/s)+(1.00 kg)(1.20 m/s)

= 1.00 kg·m/s

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• Thus, in Case 1, the momentum is conserved.

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• Is momentum conserved in Case 2 and in Case 3?

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• Next, consider the kinetic energy of the system in each of the previous cases.

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• The kinetic energy of the system before and after the collision in Case 1 is represented by the following equations:

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• In Case 1, the kinetic energy of the system increased.

• If energy in the system is conserved, then one or more of the other forms of energy must have decreased.

• Perhaps when the two carts collided, a compressed spring was released, adding kinetic energy to the system.

• This kind of collision is sometimes called a superelastic or explosive collision.

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• After the collision in Case 2, the kinetic energy is equal to:

• Kinetic energy remained the same after the collision.

• A collision in which the kinetic energy does not change, is called an elastic collision.

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• Collisions between hard, elastic objects, such as those made of steel, glass, or hard plastic, often are called nearly elastic collisions.

• After the collision in Case 3, the kinetic energy is equal to:

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• Kinetic energy decreased. Some of it may have been converted to thermal energy. This kind of collision, in which kinetic energy decreases, is called an inelastic collision.

• Objects made of soft, sticky material, such as clay, act in this way.

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• The three kinds of collisions can be represented using bar graphs, such as those shown in the figure.

• Although the kinetic energy before and after the collisions can be calculated, only the change in other forms of energy can be found.

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• In collisions, you can see how momentum and energy are really very different.

• Momentum is conserved in a collision.

• Energy is conserved only in elastic collisions.

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• It is also possible to have a collision in which nothing collides.

• If two lab carts sit motionless on a table, connected by a compressed spring, their total momentum is zero.

• If the spring is released, the carts will be forced to move away from each other.

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• The potential energy of the spring will be transformed into the kinetic energy of the carts.

• The carts will still move away from each other so that their total momentum is zero.

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• The understanding of the forms of energy and how energy flows from one form to another is one of the most useful concepts in science.

• The term energy conservation appears in everything from scientific papers to electric appliance commercials.

SECTION11.2

Write the law of conservation of energy and state the formula for the mechanical energy of the system.

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Section Check

The law of conservation of energy states that in a closed, isolated system, energy can neither be created nor destroyed; rather, energy is conserved. Under these conditions, energy changes from one form to another, while the total energy of the system remains constant.

Since energy is conserved, if no other forms of energy are present, kinetic energy and potential energy are inter-convertible.

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Section Check

Answer

Mechanical energy of a system is the sum of the kinetic energy and potential energy if no other forms of energy are present.

That is, mechanical energy of a system E = KE + PE.

SECTION11.2

Section Check

Answer

Two brothers, Jason and Jeff, of equal masses, jump from a house 3 m high. If Jason jumps to the ground and Jeff jumps to a platform 2 m high, what can you say about their kinetic energy?

A. The kinetic energy of Jason when he reaches the ground is greater than the kinetic energy of Jeff when he lands on the platform.

B. The kinetic energy of Jason when he reaches the ground is less than the kinetic energy of Jeff when he lands on the platform.

C. The kinetic energy of Jason when he reaches the ground is equal to the kinetic energy of Jeff when he lands on the platform.

D. Neither Jason nor Jeff possesses kinetic energy.

SECTION11.2

Section Check

Reason: The kinetic energy at the ground level is equal to the potential energy at the top. Since potential energy is proportional to height, and since the brothers have equal masses, Jason will have greater kinetic energy since his height above the ground decreases more than Jeff’s.

SECTION11.2

Section Check

Answer

A car of mass m1 moving with velocity v1 collides with

another car of mass m2 moving with velocity v2. After

collision, the cars lock together and move with velocity v3. Which of the following equations about their

momentums is true?

A. m1v1 + m2v2 = (m1 + m2)v3

B. m1v2 + m2v1 = (m1 + m2)v3

C. m1v1 + m2v2 = m1v2 + m2v1

D. m1v12 + m2v2

2 = (m1 + m2)v3

SECTION11.2

Section Check

Reason: By the law of conservation of momentum, we know that the total momentum before a collision is equal to the total momentum after a collision.

SECTION11.2

Section Check

Answer

Reason: Before the collision, the car of mass m1 was moving

with velocity v1 and the car of mass m2 was moving

with velocity v2. Hence, the total momentum before

the collision was ‘m1v1 + m2v2’. After the collision,

the two cars lock together. So, the mass of the two cars is (m1 + m2) and the velocity with which the two

cars are moving is v3. The total momentum after the

collision is ‘(m1 + m2)v3.’ Therefore, we can write,

m1v1 + m2v2 = (m1 + m2)v3.

SECTION11.2

Section Check

Answer

CHAPTER

Resources

Physics Online

Study Guide

Chapter Assessment Questions

Standardized Test Practice

• The translational kinetic energy of an object is proportional to its mass and the square of its velocity. The rotational kinetic energy of an object is proportional to the object’s moment of inertia and the square of its angular velocity.

SECTION11.1

Study Guide

• The gravitational potential energy of an object-Earth system depends on the object’s weight and its distance from the reference level. The reference level is the position where the gravitational potential energy is defined to be zero (h = 0).

GPE = mgh

• Elastic potential energy is energy due to stretching, compressing, or bending an object.

SECTION11.1

Study Guide

• Albert Einstein recognized that mass itself is a form of energy. This energy is called rest energy.

Eo = mc2

SECTION11.1

Study Guide

• The total energy of a closed, isolated system is constant. Within the system, energy can change form, but the total amount of energy does not change. Thus, energy is conserved.

• The sum of kinetic and potential energy is called mechanical energy.

ME = KE + PE

SECTION11.2

Study Guide

In a closed, isolated system where the only forms of energy are kinetic energy and potential energy, mechanical energy is conserved. The mechanical energy before an event is the same as the mechanical energy after the event.

KEbefore + PEbefore = KEafter + PE after

SECTION11.2

Study Guide

• Momentum is conserved in collisions if the external force is zero. The kinetic energy may be unchanged or decreased by the collision, depending on whether the collision is elastic or inelastic. The type of collision in which the kinetic energy before and after the collision is the same is called an elastic collision. The type of collision in which the kinetic energy after the collision is less than the kinetic energy before the collision is called an inelastic collision.

SECTION11.2

Study Guide

A car and a train are moving with the same velocity. Which of the following statements about their kinetic energy is true?

A. Neither the car nor the train possesses kinetic energy.

B. The kinetic energy of the car is greater than that of the train.

C. The kinetic energy of the train is greater than thatof the car.

D. Both the car and the train are running with the same kinetic energy.

CHAPTER

Chapter Assessment

Reason: Kinetic energy of an object is half times the product of its mass and the square of its velocity.

Kinetic energy is directly proportional to mass. As the velocities of the car and the train are the same, the train will have greater kinetic energy since it has greater mass.

CHAPTER

Chapter Assessment

When an arrow is thrown by hand, its speed is less than the speed when it is shot using a bow. Explain why.

CHAPTER

Chapter Assessment

Answer: When the string of a bow is pulled, work is done on the bow. Hence, the elastic potential energy is stored in the string of this bow. Before the string is released, the energy is all potential. As the string is released, the energy is transferred to the arrow as kinetic energy. When the arrow is shot using a bow, more energy is supplied to the system than when the arrow is thrown by hand. Hence, the speed of the arrow is less when it is thrown by hand.

CHAPTER

Chapter Assessment

If a car is moving backward, what can we say about its kinetic energy?

A. Kinetic energy is negative as it is moving backward.

B. Kinetic energy is positive.

C. The car does not possess kinetic energy.

D. Kinetic energy of the car gets stored as gravitational potential energy.

CHAPTER

Chapter Assessment

Reason: Kinetic energy of an object is half the product of its mass and the square of its velocity.

Kinetic energy is proportional to the square of velocity. Since the square of any number is always positive, kinetic energy is always positive.

CHAPTER

Chapter Assessment

A clock has an hour hand, minute hand, and second hand of equal masses. Which of the following statements is true?

A. The kinetic energy of the hour hand is greater than that of the minute hand and the second hand.

B. The kinetic energy of the minute hand is greater than that of the hour hand and the second hand.

C. The kinetic energy of the second hand is greater than that of the hour hand and the minute hand.

D. All the hands are moving with equal kinetic energy.

CHAPTER

Chapter Assessment

Reason: Kinetic energy is equal to one half mass times velocity squared.

The kinetic energy of an object is directly proportional to the square of the speed of that object. The masses of all the hands are the same. Therefore, the hand with the greatest speed has the greatest kinetic energy. This is the second hand.

CHAPTER

Chapter Assessment

A spring-loaded toy car is kept on the floor after compressing the spring. What form of energy does the toy car use to move?

A. stored gravitational potential energy

B. stored kinetic energy

C. linear kinetic energy

D. stored elastic potential energy

CHAPTER

Chapter Assessment

Reason: When a spring is compressed or stretched, or an object is bent, energy is stored as elastic potential energy. Hence, since the spring of the toy car is compressed, energy is stored as elastic potential energy. The toy car uses this stored elastic potential energy to move. As the toy car moves, the elastic potential energy gets converted into other forms of energy such as linear or rotational kinetic energy.

CHAPTER

Chapter Assessment

A bicyclist increases her speed from 4.0 m/s to 6.0 m/s. The combined mass of the bicycle and the bicyclist is 55 kg. How much work did the bicyclist do in increasing her speed?

A. 11 J

B. 28 J

C. 55 J

D. 550 J

CHAPTER

The illustration shows a ball swinging freely in a plane. The mass of the ball is 4.0 kg. Ignoring friction, what is the maximum speed of the ball as it swings back and forth?

A. 0.14 m/s

B. 21 m/s

C. 7.0 m/s

D. 49 m/s

CHAPTER

You lift a 4.5-kg box from the floor and place it on a shelf that is 1.5 m above the ground. How much energy did you use in lifting the box?

A. 9.0 J

B. 49 J

C. 11 J

D. 66 J

CHAPTER

You drop a 6.0×10−2 −kg ball from a height of 1.0 m above a hard, flat surface. The ball strikes the surface and loses 0.14 J of its energy. It then bounces back upward. How much kinetic energy does the ball have just after it bounces off the flat surface?

A. 0.20 J

B. 0.59 J

C. 0.45 J

D. 0.73 J

CHAPTER

You move a 2.5-kg book from a shelf that is 1.2 m above the ground to a shelf that is 2.6 m above the ground. What is the change in the book’s potential energy?

A. 1.4 J

B. 25 J

C. 3.5 J

D. 34 J

CHAPTER

Use the Process of Elimination

Test-Taking Tip

On any multiple-choice test, there are two ways to find the correct answer to each question. Either you can choose the right answer immediately or you can eliminate the answers that you know are wrong.

CHAPTER

Money Model

CHAPTER

Chapter Resources

Throwing or Catching a Ball

CHAPTER

Chapter Resources

Amount of Money in Various Forms

CHAPTER

Chapter Resources

Kinetic Energy and Potential Energy of a System

CHAPTER

Chapter Resources

The Gravitational Potential Energy of the Bowling Ball

CHAPTER

Chapter Resources

System Consisting of a 10.0-N Ball and Earth

CHAPTER

Chapter Resources

A Ball Rolling Down a Ramp

CHAPTER

Chapter Resources

Graph of the Changing Potential and Kinetic Energies of a Pendulum

CHAPTER

Chapter Resources

Energy Bar Graphs

CHAPTER

Chapter Resources

Conservation of Mechanical Energy

CHAPTER

Chapter Resources

Case 1: Two Objects Moving Apart in Opposite Directions

CHAPTER

Chapter Resources

Case 2: Moving Object Coming to Rest and the Stationary Object Beginning to Move and Case 3: Two Objects Sticking Together and Moving As One

CHAPTER

Chapter Resources

Three Kinds of Collisions

CHAPTER

Chapter Resources

Kinetic Energy

CHAPTER

Chapter Resources

A Bullet Going Through a Motionless Wooden Block

CHAPTER

Chapter Resources

A Child Sliding Down a Playground Slide

CHAPTER

Chapter Resources

Consider an elastic collision between two objects of equal mass, such as a cue ball with velocity, v, hitting a motionless billiard ball head-on.

In this case, after the collision, the cue ball is motionless and the other ball rolls off at velocity, v.

It is easy to prove that both momentum and energy are conserved in this collision.

CHAPTER

Chapter Resources

The other simple example is to consider a skater of mass, m, with velocity, v, running into another skater of equal mass who happens to be standing motionless on the ice.

If they hold on to each other after the collision, they

will slide off at a velocity of because of the

conservation of momentum.

CHAPTER

Chapter Resources

The final kinetic energy of the pair would be equal to

which is one quarter the initial

kinetic energy. This is because the collision was inelastic.

CHAPTER

Chapter Resources

Conservation of Mechanical Energy

During a hurricane, a large tree limb, with a mass of 22.0 kg and at a height of 13.3 m above the ground, falls on a roof that is 6.0 m above the ground.

A. Ignoring air resistance, find the kinetic energy of the limb when it reaches the roof.

B. What is the speed of the limb when it reaches the roof?

CHAPTER

Chapter Resources

You lift a 7.30-kg bowling ball from the storage rack and hold it up to your shoulder. The storage rack is 0.610 m above the floor and your shoulder is 1.12 m above the floor.

CHAPTER

Chapter Resources

Gravitational Potential EnergyA. When the bowling ball is at your shoulder, what is

the bowling ball’s gravitational potential energy relative to the floor?

B. What is the bowling ball’s gravitational potential energy relative to the storage rack?

C. How much work was done by gravity as you lifted the ball from the rack to shoulder level?

CHAPTER

Chapter Resources

End of Custom Shows

Chapter 11: Energy and its Conservation PHYSICS Principles and Problems

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